$\mu$-$\tau$ symmetry breaking and CP violation in the neutrino mass matrix

Abstract

The $\mu$-$\tau$ exchange symmetry in the neutrino mass matrix and its breaking as a perturbation are discussed. The exact $\mu$-$\tau$ symmetry restricts the 2-3 and 1-3 neutrino mixing angles as ${\theta }_{23}=\pi /4$ and ${\theta }_{13}=0$ at a zeroth order level. We claim that the $\mu$-$\tau$ symmetry breaking prefers a large CP violation to realize the observed value of ${\theta }_{13}$ and to keep ${\theta }_{23}$ nearly maximal, though an artificial choice of the $\mu$-$\tau$ breaking can tune ${\theta }_{23}$, irrespective of the CP phase. We exhibit several relations among the deviation of ${\theta }_{23}$ from $\pi /4$, ${\theta }_{13}$ and Dirac CP phase $\delta$, which are useful to test the $\mu$-$\tau$ breaking models in the near future experiments. We also propose a concrete model to break the $\mu$-$\tau$ exchange symmetry spontaneously and its breaking is mediated by the gauge interactions radiatively in the framework of the extended gauge model with $B-L$ and $L_{\mu }-L_{\tau }$ symmetries. As a result of the gauge mediated $\mu$-$\tau$ breaking in the neutrino mass matrix, the artificial choice is unlikely, and a large Dirac CP phase is preferable.

Comments: 25 pages